A hint in the book told us to prove that $p_k^{a_k}$ divides $y$. However, I do not have any idea how to do that.
Also, even if I managed to prove that, I don't see how that may lead to the final result.

1 Answer
1

Here is a general sketch of how to proceed. Your equation is expressible as
$$x^2\prod_{p_i}\left(1-\frac{1}{p_i}\right) = y^2\prod_{q_i}\left(1-\frac{1}{q_i}\right)$$
If you clear denominators and cancel a few things, then you end up with
$$\left(\prod_{p_i}p_i^{2a_i - 1}\right)\left(\prod_{p_i}(p_i-1)\right)=\left(\prod_{q_i}q_i^{2b_i - 1}\right)\left(\prod_{q_i}(q_i-1)\right)$$
Show that the largest prime factor in the two expressions occurs in
$$\left(\prod_{p_i}p_i^{2a_i - 1}\right)\ \ \text{and}\ \ \left(\prod_{q_i}q_i^{2b_i - 1}\right)$$
respectively. This allows you to conclude that the largest prime factor of $x$ and $y$ are equal. The result follows inductively.